Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution

TL;DR

The paper investigates Ramanujan-type congruences for PEND and POND partitions modulo $3$ and reveals a precise link between their multiplicities via an Atkin–Lehner involution. By constructing extended generating functions and applying the $U_3^{2\alpha+1}$ operator, the authors place these series in $\mathcal{M}(\Gamma_0(12))$ and show that an Atkin–Lehner involution maps one family to the other, up to explicit sign and scalar factors. They further establish an explicit Hauptmodul-based isomorphism between the associated modular objects, demonstrating that knowledge of one congruence family immediately yields the other. The work highlights how congruence multiplicities encode deeper arithmetic information and suggests analogous connections for PED/POD partitions, offering a unified modular-form framework to transfer congruences across parity-restricted partition families.

Abstract

For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary $q$-series identities and generating function manipulations, along with mathematical induction: For all $α\geq 1$ and all $n\geq 0,$ $$\mathrm{pend}\left(3^{2α+1}n+\frac{17\cdot 3^{2α}-1}{8}\right) \equiv 0 \pmod{3}, \textrm{ and}$$ $$\mathrm{pond}\left(3^{2α+1}n+\frac{23\cdot 3^{2α}+1}{8}\right) \equiv 0 \pmod{3},$$ where $\mathrm{pend}(n)$ counts the number of PEND partitions of weight $n$ and $\mathrm{pond}(n)$ counts the number of POND partitions of weight $n$. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin--Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 3.1